\documentclass[
  fontsize=11pt,
  paper=a4,
  parskip=half,
  enlargefirstpage=on,    % More space on first page
  fromalign=right,        % PLacement of name in letter head
  fromphone=on,           % Turn on phone number of sender
  fromemail=on,
  fromrule=aftername,     % Rule after sender name in letter head
  addrfield=on,           % Adress field for envelope with window
  backaddress=off,         % Sender address in this window
  subject=beforeopening,  % Placement of subject
  locfield=narrow,        % Additional field for sender
  foldmarks=off,           % Print foldmarks
  sections,
]{scrlttr2}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{amsmath}
\usepackage{MnSymbol}

\setkomafont{fromname}{\sffamily \LARGE}
\setkomafont{fromaddress}{\sffamily}%% statt \small
\setkomafont{pagenumber}{\sffamily}
\setkomafont{subject}{\bfseries}
\setkomafont{backaddress}{\mdseries}

\LoadLetterOption{DIN}
\setkomavar{fromname}{Vladimír Duša}

\setkomavar{fromaddress}{KTIML MFF\\Malostranské náměstí 25\\118 00 Praha 1, Czech Republic}
\setkomavar{fromphone}{+420 723 753 853}
\setkomavar{fromemail}{dusa@ktiml.mff.cuni.cz}
\setkomavar{backaddressseparator}{\enspace\textperiodcentered\enspace}
\setkomavar{signature}{Vladimír Duša}
\setkomavar{place}{Praha}
\setkomavar{date}{\today}
\setkomavar{enclseparator}{: }

\newcommand{\reaction}[1]{\\\textbf{Reaction:} #1}

\newcommand{\act}{\mathcal{O}}
\newcommand{\eqtree}{\mathcal{T}}
\newcommand{\eqts}[1]{\eqtree^{#1}}
\newcommand{\eqt}{\eqts{\sigma}}
\newcommand{\eqg}{G_{\sigma}^{=}}
\newcommand{\eqtforw}[1]{\overset{\twoheadrightarrow}{\rule{0pt}{1.2ex}\smash{\eqt}}(#1)}

\begin{document}
  \begin{letter}{To The Journal of Scheduling Editorial Office}
    \setkomavar{subject}{Comments to the reports on the paper \#JOSH-D-13-00057}
    \opening{Ladies and Gentlemen,}

    we would like to thank you for the reports on our paper \emph{Scheduling with Convex Cost Functions and Temporal Constraint}. We have considered all your comments. Some of them were very useful and their implementation improved the quality of the paper. Please follow the remarks below describing all improvements made in the paper. Each point consists of your comment followed by our reaction.

\section{Our comments to the first review}
    \begin{itemize}    	
    	\item page 2, line 25, right: earliness and tardiness penalties.
    	\reaction{Typo has been corrected.}
    	
    	\item page 3, line 26, right: in the graph union definition, we must have $V_1 \cup V_2 = \emptyset$.
    	\reaction{Implemented.}
    	
    	\item page 3, line 31, right: is the subgraph of $G$ induced on $\widetilde{V}$ (given $\widetilde{V}$, $\widetilde{G}(\widetilde{V}$) is unique!)
    	\reaction{Implemented.}
    	
    	\item page 5, line 3, left: Let $\eqt$ be an induced subgraph of $\eqg$
    	\reaction{The equality graph needn't necessarily be a tree. Thus the equality tree can't be defined as an induced subgraph, but just as a plain subgraph. Let us for example consider following equality graph $$\eqg=(\{i,j,k,l\},\{(i,j), (i,k), (j,l),(k,l)\}).$$ Since $\eqt$ is defined as a tree (orientation of edges is ignored), also $$\eqt=(\{i,j,k,l\}, \{(i,j), (i,k), (j,l)\})$$ is a possible equality tree. The term \emph{induced subgraph} is used in the paper in the definition of the \emph{graph subtraction} and \emph{backward/forward closure}}.
    	
    	\item page 5, line 26, left: it could be mentioned that the subgraphs of $\eqg$ induced by singletons are active equality trees.
    	\reaction{This remark has been inserted into the text.}
    	
    	\item page 6, line 40, left: Let $G = (V,E)$
  	    \reaction{Typesetting has been corrected.}
    	
    	\item page 6, line 59, left: (resp. $\eqtforw{B}$ is late).
		\reaction{The definition is correct. It says that the backward closure $\overleftarrow{\eqt}(B)$ is early and the forward closure $\overrightarrow{\eqt}(F)$ is late. It would be possible to use just one subset of $\act_{\eqt}$ instead of two subsets $F$ and $B$ in the definition, but we believe, that our way is better understandable.}
    	
    	\item page 6, line 29, right: $j = \widetilde{j}$ is not possible
    	\reaction{We have changed the text: Let us consider two different temporal constraints $(i,j), (\widetilde{i},\widetilde{j})\in P$.}
    	
    	\item page 6, line 52, right: we call\ldots
    	\reaction{Typo has been corrected.}
    	
    	\item page 7, line 38, left: of an on time active equality forest\ldots
    	\reaction{Implemented.}
    	
    	\item page 7, line 53, left: you should use an another term than "shifted forward" since this action is far more complex than the basic shift of Definition 5. Moreover what happens if $i$ and $j$ are in the same equality tree? 
    	\reaction{Term \emph{shift} in the sense of the complex action has been replaced by the term \emph{push} in whole text describing the algorithm. Also the informal description of the algorithm has been improved.}
    	
    	\item page 8, line 8, left: if $i\notin S^{\sigma}$ or $j\notin S^{\sigma}$
		\reaction{Corrected.} 
		
    	\item However, even if the proposed algorithm seems to work, the technical part of the correctness proof is not sufficiently convincing from a mathematical point of view and should be better formalized (for example, what happens if the added violated temporal constraint (i; j) is such that i and j are initially in the same active equality tree, is not clear)
    	\reaction{Proof of the algorithm correctness has been added to the paper. See section 3.3.1.}
    	
    	\item Moreover, since the algorithm has been implemented in JAVA, a complete description of of how it works on a well chosen instance could be interesting and helpful to understand the most complex cases.
    	\reaction{The existing Java code is mentioned just for purposes of a readers interest in the implementation of the algorithm. Since we support open source projects, the code is released freely for reuse and further development. A small example describing all important steps of the algorithm is introduced in the paper.}		
    \end{itemize}

\section{Our comments to the second review}
	\begin{itemize}
		\item The authors do not specify if it is a single machine or parallel machine problem and assume implicitly that the readers are familiar with reference [3].
		\reaction{The paper defines the problem as a plain scheduling of resources independen operations (In a same way as for example Critical Path Method). All necessary information are mentioned in our paper and the reader don't need to be familiar with reference [3].}
		
		\item I am not convinced of the correctness for their claim that they have proven polynomial time solvability.
		\reaction{We rechecked the correctness of the polynomial time solvability of the problem and we didn't find any mistake. We would like to ask you for a more formal disproof of our claim.}
		
		\item In essence, I believe that a special case of their problem can be described by \ldots | strong-in tree, $p_i\geq 0$|\ldots described in Dror, Kubiak, and J.Y.-T. Leung, (1999), Tree precedence in scheduling: The strong-weak distinction, \textit{IPL} 71, pp. 127-134.
		\reaction{We studied the paper \emph{Dror, Kubiak, and J.Y.-T. Leung, (1999), Tree precedence in scheduling: The strong-weak distinction, \textit{IPL} 71, pp. 127-134}. At the first sight, both problems seem to consider the similar topic, but they are actually different.}
	\end{itemize}

\section{Another changes in the paper}
Since the topic discussed in the paper is quite complex and both reviewers encountered difficulties in understanding of some parts of the text, we implemented another changes bringing, in our point of view, better readability of the text.

\begin{itemize}
	\item The definition of the problem adopted from reference [3] and extended by the maximal distance between operations was transformed into the STN in the original text. Newly, this transformation has been removed and the problem is proposed directly as STN.
	
	\item In order to let the reader go through the paper more fluently, several bridging of the text summarizing the read section and preparing the reader for following section were inserted into the text.
\end{itemize}

We believe that all mentioned modifications improved the paper enough to be published in Journal of Scheduling.

    \closing{Sincerely}
  \end{letter}
\end{document}